Phase diagram of the dissipative Hofstadter model

A quantum particle moving in a uniform magnetic field and periodic potential (the Hofstadter model) has an energy band structure which varies in a discontinuous fashion as a function of the magnetic flux per lattice unit cell. In a real system, randomness of various kinds should "smooth out" this behavior in some way. To explore how this happens, we have studied the dissipative quantum mechanics of the Hofstadter model. We find, by virtue of a duality in a two-dimensional space parametrized by the dissipation constant and the magnetic field strength, that there are an infinite number of phase transition lines, whose density grows without limit as the dissipation goes to zero and the model reduces to the original Hofstadter model. The measurable quantity of greatest interest, the mobility, can be determined exactly in most of parameter space. The critical theory on the phase transition lines has yet to be characterized in any detail, but it has reparametrization invariance and defines a set of nontrivial backgrounds for open string theory.